DLMF:20.8.E1 (Q6830): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11945 / rank
 
Normal rank
Property / Symbols used: Q11945 / qualifier
 
Defining formula:

θ j ⁡ ( z , q ) Jacobi-theta 𝑗 𝑧 𝑞 {\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)}}

\Jacobithetaq{\NVar{j}}@{\NVar{z}}{\NVar{q}}
Property / Symbols used: Q11945 / qualifier
 
xml-id: C20.S2.SS1.m2adec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2adec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2adec
Property / Symbols used
 
Property / Symbols used: Q11947 / rank
 
Normal rank
Property / Symbols used: Q11947 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11947 / qualifier
 
xml-id: C20.S1.XMD2.m1dec
Property / Symbols used
 
Property / Symbols used: Q11948 / rank
 
Normal rank
Property / Symbols used: Q11948 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11948 / qualifier
 
xml-id: C20.S1.XMD3.m1dec
Property / Symbols used
 
Property / Symbols used: Q11950 / rank
 
Normal rank
Property / Symbols used: Q11950 / qualifier
 
Defining formula:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q11950 / qualifier
 
xml-id: C20.S1.XMD5.m1dec

Latest revision as of 14:41, 2 January 2020

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DLMF:20.8.E1
No description defined

    Statements

    Defining formula
    θ 2 ⁡ ( 0 , q ) ⁢ θ 3 ⁡ ( z , q ) ⁢ θ 4 ⁡ ( z , q ) θ 2 ⁡ ( z , q ) = 2 ⁢ ∑ n = - ∞ ∞ ( - 1 ) n ⁢ q n 2 ⁢ e i ⁢ 2 ⁢ n ⁢ z q - n ⁢ e - i ⁢ z + q n ⁢ e i ⁢ z . Jacobi-theta 2 0 𝑞 Jacobi-theta 3 𝑧 𝑞 Jacobi-theta 4 𝑧 𝑞 Jacobi-theta 2 𝑧 𝑞 2 superscript subscript 𝑛 superscript 1 𝑛 superscript 𝑞 superscript 𝑛 2 superscript 𝑒 𝑖 2 𝑛 𝑧 superscript 𝑞 𝑛 superscript 𝑒 𝑖 𝑧 superscript 𝑞 𝑛 superscript 𝑒 𝑖 𝑧 {\displaystyle{\displaystyle\frac{\theta_{2}\left(0,q\right)\theta_{3}\left(z,% q\right)\theta_{4}\left(z,q\right)}{\theta_{2}\left(z,q\right)}=2\sum_{n=-% \infty}^{\infty}\frac{(-1)^{n}q^{n^{2}}e^{i2nz}}{q^{-n}e^{-iz}+q^{n}e^{iz}}.}}
    0 references
    DLMF id
    DLMF:20.8.E1
    0 references
    Symbols used
    Q11945
    Defining formula
    θ j ⁡ ( z , q ) Jacobi-theta 𝑗 𝑧 𝑞 {\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)}}
    xml-id
    C20.S2.SS1.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q11947
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C20.S1.XMD2.m1dec
    0 references
    Q11948
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C20.S1.XMD3.m1dec
    0 references
    Q11950
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C20.S1.XMD5.m1dec
    0 references
     
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